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BREAKDOWN OF THREE DIMENSIONAL
EFFECTIVE MASS APPROXIMATION IN NARROW SUPERLATTICES
C. Priester, G. Allan, M. Lannoo
To cite this version:
C. Priester, G. Allan, M. Lannoo. BREAKDOWN OF THREE DIMENSIONAL EFFECTIVE MASS APPROXIMATION IN NARROW SUPERLATTICES. Journal de Physique Colloques, 1984, 45 (C5), pp.C5-519-C5-523. �10.1051/jphyscol:1984577�. �jpa-00224198�
JOURNAL DE PHYSIQUE
Colloque C5, suppl6ment au n04, Tome 45, a v r i l 1984 page ~ 5 - 5 1 9
BREAKDOWN OF THREE DIMENSIONAL EFFECTIVE MASS APPROXIMATION I N NARROW SUPERLATTICES
C. Priester, G. Allan and M. Lannoo
Institut Sup&rieur dlEZectronique du Nord, 3 rue Franpois Bags, 59046 LiZZe Cedex, France
Resume - Pour des superr@seaux t r e s e t r o i t s , une t h e o r i e u t i l i s a n t l ' a p p r o x i - mation de l a masse e f f e c t i v e 3 t r o i s dimensions n ' e s t p l u s valable. Nous avons Gtudie a p a r t i r de quand c e t t e t h e o r i e f a i t defaut, e t quel e s t l ' o r d r e de grandeur de l ' e r r e u r correspondante. Nous presentons egalement un t r a i t e m e n t bidimensionnel v a l i d e pour un p u i t s de 1 a 15 plans. Ce t r a i t e m e n t permet de c a l c u l e r l ' e n e r g i e de l i a i s o n par r a p p o r t a l l @ n e r g i e "exacte" du bas de l a bande. I 1 s ' a p p l i q u e egalement b i e n 3 chacune des sous-bandes.
A b s t r a c t - I n t h e l i m i t o f v e r y narrow s u p e r l a t t i c e s , a theory u s i n g t h r e e dimensional e f f e c t i v e mass approximation becomes inadequate. We have s t u d i e d a t which stage t h i s approximation becomes poor and what i s t h e magnitude o f t h e corresponding e r r o r . We a l s o present a two dimensional treatment which proves t o be v a l i d f o r a number o f planes v a r y i n g from 1 t o about 15. T h i s treatment a l l o w s t h e c a l c u l a t i o n o f the b i n d i n g energy w i t h respect t o t h e
"exact" energy o f t h e bottom o f t h e corresponding subband. I t a p p l i e s e q u a l l y we1 1 t o each o f t h e subbands.
Since the development o f Molecular Beam Epitaxy, t h e r e i s a g r e a t i n t e r e s t i n semi- conductor s u p e r l a t t i c e s , f i r s t r e a l i z e d by Esaki and Tsu [ 1 1. Many a b s o r p t i o n and luminescence experiments have been performed [ 2,3 ] and e x c i t o n s have been observed A simple and a n a l y t i c a l t h e o r y o f these systems has been proposed by Bastard [41 u s i n g a t h r e e dimensional e f f e c t i v e mass approximation, 3D E.M.A. ( r e f i n e d treatment has been proposed r e c e n t l y i n r e f . [51). This t h e o r y i s based on t h e f o l l o w i n g assumptions : i ) the d i s c o n t i n u i t i e s i n p o t e n t i a l are assumed t o be l a r g e so t h a t t h e study i s l i m i t e d t o an i s o l a t e d quantum w e l l i i ) t h e quantum w e l l i s simulated by an i n f i n i t e square w e l l , t h e e f f e c t i v e k i n e t i c energy o p e r a t o r being obtained from t h e t h r e e dimensional E.M.A. i i i ) t h e i m p u r i t y p o t e n t i a l i s superposed t o the constant p o t e n t i a l i n t h e quantum w e l l .
I t i s c l e a r t h a t such a treatment i s v a l i d f o r a l a r g e number o f planes i n t h e quantum w e l l . I t should become inadequate 1 6 ] i n t h e l i m i t o f a few planes since t h e p o t e n t i a l does no more f u l f i l l t h e c r i t e r i o n o f being s l o w l y varying.
However i t i s important t o know a t which stage these approximations become poor and what i s t h e magnitude o f t h e corresponding e r r o r . The present work t t i e s t o answer such questions. For t h i s an "exact" t e s t c a l c u l a t i o n i s f i r s t performed, f o r small number o f planes, i n a simple case o f a t i g h t b i n d i n g s band w i t h a Coulombic i m p u r i t y . We a l s o present a two dimensional E.M.A. which i s t r a c t a b l e o n l y f o r very narrow s u p e r l a t t i c e s and compare i t t o t h e exact r e s u l t s . We then present a more u s e f u l two dimensional treatment which proves t o be v a l i d i n t h e i n t e r m e d i a t e case,
i . e f o r a number o f planes ranging from 1 t o about 15. T h i s treatment represents i n f a c t t h e c e n t r a l r e s u l t o f t h i s paper s i n c e i t a l l o w s c a l c u l a t i o n o f t h e b i n d i n g energy w i t h r e s p e c t t o t h e "exact" energy o f t h e bottom o f t h e corresponding subband, which i s n o t t h e case i n B a s t a r d ' s model. F i n a l l y we discuss t h e d a l i d i t y o f E.M.A. f o r t h a t problem and show t h a t our two-dimensional treatment propides t h e n a t u r a l approximation method f o r moderately narrow and narrow s u p e r l a t t i c e s .
Here we want t o perform an "exact" numerical t e s t c a l c u l a t i o n . For t h i s we consider an i n f i n i t e quantumwell, w i t h a c e n t r a l donor i m p u r i t y . I n t h e w e l l , we have 1,3 o r
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984577
C5-520 JOURNAL DE PHYSIQUE
3 p l a n e s o f atoms. We o b t i i n a two dimensional i m p u r i t y problem t h a t can be s o l v e d i n t i g h t b i n d i n g u s i n g a numerical t e c h n i q u e based on Green's f u n c t i o n s t h e o r y [ 5 I . We c a n a l s o t r e a t t h i s problem w i t h t h e u s u a l t e c h n i q u e s o f e f f e c t i v e mass t h e o r y
[ 7 1 .
The r e s u l t s o b t a i n e d a r e r e p o r t e d i n T a b l e 1 showing v e r y good agreement between t h e two c a l c u l a t i o n s . T h i s shows t h e n u s e f u l n e s s o f 2DE.M.A. f o r t h a t problem.
a (one plane) b (three planes)
T a b l e 1 : D i s t a n c e between t h e bottom o f t h e band and t h e i m p u r i t y s t a t e , i n a t o m i c u n i t s , o b t a i n e d u s i n g a numerical c a l c u l a t i o n (LCAO, on an s c u b i c , u s i n g r e c u r s i o n ) and u s i n g 20 E.M.A.
a ) system made by one p l a c e
b) system made by t h r e e planes ( a n t i s y m m e t r i c a l s o l u t i o n s ) .
s t a t e
=la
E 2 ~
'28
The t r e a t m e n t developped j u s t b e f o r e i s a d i s c r e t e method t h a t can t r e a t o n l y v e r y few p l a n e s ( N < 3 ) . I t i s t h u s o f c o n s i d e r a b l e i n t e r e s t t o d e v i s e a more g e n e r a l method t o reduce t h e problem t o a 2D e f f e c t i v e mass e q u a t i o n . We s h a l l adapt f o r t h i s t h e g e n e r a l method which a l l o w s t o d e r i v e 30 E.M.A. i n k space.
We have proved t h a t 2D E.M.A. i s a p p l i c a b l e as l o n g as i m p u r i t y b i n d i n g e n e r g i e s a r e s m a l l e r t h a n t h e energy s p l i t t i n g between t h e subbands. I f we t a k e f o r t h e r a t i c o f t h e s e two e n e r g i e s a l i m i t i n g v a l u e o f 0.25 t h e n we f i n d t h a t o u r 2D t r e a t m e n t should be v a l i d f o r N 6 10 t o 15.
"Exact calculation"/2~ E.U.A.
0.00419/0.0040
0.000441/0.000444
0.000448/0.000444
The i m p o r t a n t p o i n t about t h e 2D E.M.A. i s t h a t i t c o n t a i n s an e f f e c t i v e i m p u r i t y p o t e n t i a l Un(r ) which i s an average o f V(Q/,Z) o v e r z. T h i s can
be performed i f we know t h e s o l u t i o n o f Ho, t h e h a m i l t o n i e n o f t h e s u p e r l a t t i c e i n t h e absence o f t h e i m p u r i t y .
"Exact calculation/2~ E.M.A.
0.00259/0.00249
0.000434/0.000417
0.000372/0.000379
The r e s u l t s o f t h i s c a l c u l a t i o n a r e p l o t t e d on f i g u r e 1 where t h e y a r e compared t o t h o s e o f t h e d i s c r e t e LCAO c a l c u l a t i o n s . We can observe t h a t t h e r e i s s i g n i f i c a n t d i f f e r e n c e o n l y f o r v e r y narrow s u p e r l a t t i c e s ( t h e maximum e r r o r , f o r N = 1, r e p r e s e n t s o n l y 13% o f t h e b i n d i n g energy, w h i l e f o r N > 1 i t i s p r a c t i c a l l y n e g l i g i b l e ) . The 2D E.M.A. i n i t s c o n t i n u o u s v e r s i o n t h u s r e p r e s e n t s a q u i t e a c c u r a t e method f o r p r e d i c t i n g t h e b i n d i n g e n e r g i e s f o r m o d e r a t e l y s m a l l number o f planes (N 6 10 t o 1 5 ) . As we have shown, f o r v e r y narrow s u p e r l a t t i c e s i t can be reduced t o i t s d i s c r e t e v e r s i o n which g i v e s e s s e n t i a l l y " e x a c t " r e s u l t s .
F i g u r e 1 : Comparison o f b i n d i n g e n e r g i e s o b t a i n e d by a c o n t i n u o u s model ( c u r v e ) and a d i s c r e t e c a l c u l a t i o n ( c r o s s e )
(Energy i s reduced energy : E/(mxe / 2 ~ 2 f f i ) ) . 2
We now compare o u r r e s u l t s w i t h t h o s e o f p r e v i o u s c a l c u l a t i o n s and m a i n l y t h e one by B a s t a r d [41 based on t h e u s e o f 3D E.M.A.. T h i s work was m a i n l y concerned by t h e l o w e r i m p u r i t y s t a t e s d e r i v e d f r o m t h e l o w e r subband. B a s t a r d ' s model i n t r o d u c e s two l e v e l s o f a p p r o x i m a t i o n : one on €1, t h e fundamental state,and a n o t h e r one on AEls, t h e b i n d i n g energy. The f i r s t one corresponds t o t h e f a c t t h a t ~1 i s c e r t a i n l y a v e r y c r u d e a p p r o x i m a t i o n t o t h e e x a c t b o t t o m o f t h e l o w e r band f o r v e r y narrow s u p e r l a t t i c e s and t h i s e r r o r i s n o t easy t o e s t i m a t e i n a g e n e r a l manner. I f we now l o o k a t t h e b i n d i n g energy we f i n d t h a t , t h e r e , t h e s i t u a t i o n i s much more encoura- g i n g . We have p l o t t e d on f i g u r e 2 t h e AElS o f B a s t a r d ' s 3D model and o f o u r 2D one and f i n d t h a t t h e y g i v e p r a c t i c a l l y t h e same answer f o r N < 20. A f t e r t h a t t h e 3D r e s u l t i s lower, meaning t h a t t h e 3D wave f u n c t i o n i s b e t t e r f o r l a r g e s u p e r l a t t i c e s which i s q u i t e normal.
A f i r s t c o n c l u s i o n t h a t emerges i s t h a t t h e b i n d i n g energy o f t h e 1s s t a t e p r e d i c t e d by t h e 3D E.Y.A. i s e x t r e m e l y good. Thus i t s m a i n d e f i c i e n c y i s t h a t i t i s n o t r e l a t e d t o t h e e x a c t bottom o f t h e band b u t r a t h e r t o an approximate v a l u e which can i n t r o d u c e e a s i l y e r r o r s o f t h e o r d e r o f t h e b i n d i n g energy. I n t h i s r e g a r d 20 E.M.A. as d e r i v e d h e r e i s s u p e r i o r s i n c e i t g i v e s a b i n d i n g energy w i t h r e s p e c t t o t h e e x a c t bottom o f t h e bands. Another s t r o n g i n t e r e s t o f t h e 2D E.M.A. t r e a t m e n t i s t h a t i t a p p l i e s e q u a l l y w e l l t o each o f t h e subbands, which i s e v i d e n t l y n o t t r u e o f B a s t a r d ' s model (where t h e r e i s no s y s t e m a t i c way o f b u i l d i n g wave f u n c t i o n s f o r i m p u r i t y s t a t e s d e r i v e d f r o m e x c i t e d subbands which a r e o r t h o g o n a l t o t h e l o w e r s t a t e s ) . We can c o n s i d e r t h e d e n s i t y o f s t a t e s o f t h e p e r f e c t s u p e r l a t t i c e as t h e s u p e r p o s i t i o n o f b i d i m e n s i o n a l subbands. We have shown t h a t , t o a v e r y good a p p r o x i m a t i o n , t h e l o w e r s t a t e under t h e n t h subband i s g i v e n w i t h i n t h e bidimen- s i o n a l e f f e c t i v e mass a p p r o x i m a t i o n by
-t + 2 2 a2 L / 2 + -f
- f i a
[E-E ( k = O)]F ( r ) = [-(- + --2)+ / s i n ( k z+
n // '1 2mX ax2 ay -L/Z n V ( r / 5 ~ ) d z l F n ( r / / 1
JOURNAL DE PHYSIQUE
E i g u r e 2 : B i n d i n g energy o f t h e l o w e r s t a t e v e r s u s L, t h e w i d t h o f t h e w e l l (expressed i n i n t e r a t o m i c d i s t a n c e s ) , o b t a i n e d u s i n g 2D (dashed l i n e ) and %D ( d o $ t d l i n e ) c o n t i n u o u s models. (Energy i s reduced energy : E/(m e2/2b K 5 ) ) .
-a r
W i t h a t r i a l f u n c t i o n o f t h e f o r m A, I . The i m p u r i t y energy l e v e l s o b t a i n e d by m i n i m i z a t i o n a r e r e p o r t e d i n T a b l e 2.
T a b l e I1 - B i n d i n g energy ( i n a.u.) o f t h e l o w e s t i m p u r i t y energy l e v e l s under t h e f i v e f i r s t subbands, f o r d i f f e r e n t w i d t h s o f l a y e r s (a= 5 a.u.,
ms= 0 . 2 a.u, K = 1 0 ) . F o r l a r g e v a l u e s o f L t h e model i s no l o n g e r v a l i d .
The c a l c u l a t i o n j u s t d e s c r i b e d assumes t h a t each i m p u r i t y s t a t e d e r i v e s o n l y f r o m one b i d i m e n s i o n a l subband. T h i s w i l l remain t r u e as l o n g i t s b i n d i n g energy i s much s m a l l e r t h a n t h e d i s t a n c e i n energy between i t s subband and t h e l o w e r one. Nhen t h i s c o n d i t i o n i s r e a l i z e d , o u r two-dimensional s t a t e s a r e n e a r l y e x a c t e i g e n s t a t e s
o f t h e problem. They can t h u s be used t o s t u d y t h e c o u p l i n g t o o t h e r subbands g i v i n g us i n f o r m a t i o n a b o u t t h e s h i f t and broadening o f t h e c o r r e s p o n d i n g resonances.
The numerical r e s u l t s o b t a i n e d f o r ER and r a l l o w t o conclude t h a t t h e c o u p l i n g i s v e r y weak ( b o t h s h i f t and broadening p r e s e n t s m a l l v a l u e s i n r e g a r d t o t h e b i n d i n g energy E B - E ~ t h a t c h a r a c t e r i z e s t h e s t a t e ) . T h i s r e s u l t a l s o a p p l i e s t o i m p u r i t y s t a t e s d e r i v e d f r o m h i g h e r subbands s i n c e t h e q u a n t i t y Eo-E w i l l i n c r e a s e l e a d i n g t o weaker c o u p l i n g and s h a r p e r resonances. Thus we can safeBy c o n c l u d e t h a t i n a l l cases t h e c o u p l i n g c o r r e c t i o n s can be n e g l e c t e d .
C o n c l u s i o n
We have d e s c r i b e d i n t h i s work d i f f e r e n t approaches t o t h e c a l c u l a t i o n of i m p u r i t y s t a t e s i n s u p e r l a t t i c e s based on t h e e f f e c t i v e mass a p p r o x i m a t i o n . We have f i r s t performed an e x a c t numerical t e s t c a l c u l a t i o n f o r a s i n g l e band and a narrow super- l a t t i c e . We have t h e n proposed a new 2D e f f e c t i v e mass scheme, v a l i d f o r moderate and s m a l l number o f p l a n e s N ,< 15, b u t a p p l i c a b l e t o i m p u r i t y s t a t e s d e r i v e d f r o m a l l subbands. We have compared i t t o B a s t a r d ' s 30 E.M.A., a p p l i c a b l e t o t h e l o w e r s t a t e s o n l y , showing t h a t b o t h schemes l e a d t o p r a c t i c a l l y i d e n t i c a l b i n d i n g e n e r g i e s i n t h e i n t e r m e d i a t e range (3 < N ,c 15). F i n a l l y we have c a l c u l a t e d t h e c h a r a c t e r i s t i c s o f i m p u r i t y l e v e l s d e r i v e d from h i g h e r o r d e r subbands i n quantum l a y e r s t r u c t u r e s . Such s t a t e s f a l l i n t h e continuum o f t h e l o w e r subbands and t h u s become r e s o n a n t s t a t e s . Using v a r i a t i o n n a l l y determined wave f u n c t i o n s we have c a l c u l a t e d t h e energy and w i d t h o f t h e s e resonances o b t a i n i n g s m a l l numerical v a l u e s . T h i s c o n f i r m s t h e e x p e r i m e n t a l f a c t t h a t narrow e x c i t o n i c s t a t e s have been observed f o r such h i g h e r o r d e r subbands.
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